Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{3z^2 - 18z - 48}{z^3 - 3z^2 - 40z}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {3(z^2 - 6z - 16)} {z(z^2 - 3z - 40)} $ $ a = \dfrac{3}{z} \cdot \dfrac{z^2 - 6z - 16}{z^2 - 3z - 40} $ Next factor the numerator and denominator. $ a = \dfrac{3}{z} \cdot \dfrac{(z - 8)(z + 2)}{(z - 8)(z + 5)}$ Assuming $z \neq 8$ , we can cancel the $z - 8$ $ a = \dfrac{3}{z} \cdot \dfrac{z + 2}{z + 5}$ Therefore: $ a = \dfrac{ 3(z + 2)}{ z(z + 5)}$, $z \neq 8$